Linked Questions

24 votes
6 answers
4k views

Hamiltonian for relativistic free particle is zero

One possible Lagrangian for a point particle moving in (possibly curved) spacetime is $$L = -m \sqrt{-g_{\mu\nu} \dot{x}^\mu \dot{x}^\nu},$$ where a dot is a derivative with respect to a parameter $\...
Javier's user avatar
  • 28.6k
16 votes
5 answers
7k views

Why can't any term which is added to the Lagrangian be written as a total derivative (or divergence)?

All right, I know there must be an elementary proof of this, but I am not sure why I never came across it before. Adding a total time derivative to the Lagrangian (or a 4D divergence of some 4 ...
David Santo Pietro's user avatar
15 votes
3 answers
12k views

Energy-momentum tensor from Noether's theorem

In the book "Quantum Field Theory" by Itzykson and Zuber the following derivation for the stress-energy tensor is proposed (p. 22): Assume a Lagrangian density depending on the spacetime ...
Whelp's user avatar
  • 4,156
11 votes
2 answers
3k views

Why is the Hamiltonian zero in relativity?

I'm trying to understand something with the lagrangian and the hamiltonian formalisms in relativity theory, and why the following result cannot be the same in classical (non-relativistic) mechanics. ...
Cham's user avatar
  • 7,677
17 votes
1 answer
5k views

Trick for deriving the stress tensor in any theory

In D. Tong's notes on string theory (pdf) section 4.1.1 he explains a trick for deriving the stress-energy tensor which arises from translations in the base manifold of the field theory (in this case ...
Prastt's user avatar
  • 929
8 votes
1 answer
2k views

How to calculate an axial anomaly in 1+1 dimensions?

As far as I understand, an axial $U(1)$ transformation transforms a two-component spinor like $$ \psi \to \psi'=\text e^{\text i\epsilon \gamma^5 }\psi,\qquad \psi=\begin{pmatrix}\psi_1\\\psi_2\end{...
ersbygre1's user avatar
  • 2,678
4 votes
2 answers
5k views

Prove energy conservation using Noether's theorem

I wonder how you prove that energy is conserved under a time translation using Noether's theorem. I've tried myself but without success. What I've come up with so far is that I start by inducing the ...
Turbotanten's user avatar
  • 1,155
7 votes
3 answers
1k views

Derivation of the Noether current

(Cf. Di Francesco et al, Conformal Field Theory, pp. 40-41) I am trying to derive eqn. (2.142) or $$\delta S = \int d^d x ~\omega_a~\partial_{\mu}j^{\mu}_a \tag{2.142}$$ in the book CFT by Di ...
CAF's user avatar
  • 3,599
2 votes
2 answers
874 views

Variation of the Lagrangian and the Noether current

In Schwartz’s book, QFT and Standard Model, section 8.3.1, he writes if we then let $\alpha$ be a function of $x$, the transformed $\mathcal L_0$ can only depend on $\partial_\mu \alpha$. Thus, for ...
Roland's user avatar
  • 49
3 votes
1 answer
2k views

Noether's Theorem in Classical Field theory Confusion

Consider $N$ independent scalar fields $φ_i (x)$ in 4D space. Also consider a lagrangian density $$\mathcal{L} = \mathcal{L}(φ_i, \partial_μφ_i).$$ Suppose we perform the following infinitesimal ...
Arbiter's user avatar
  • 354
3 votes
2 answers
306 views

Variation of action in terms of energy-momentum tensor

In Di Francesco, Mathieu, and Sénéchanl, Conformal Field theory section 4.2.2 it is stated that under an arbitrary diffeomorphism $x\rightarrow x+\epsilon$ the action transforms like $$\delta S=\int d^...
Ivan Burbano's user avatar
  • 3,975
1 vote
3 answers
1k views

How did we find the Noether current $j^\mu(x) = \bar\psi(x)\gamma^\mu\psi(x)$ for Dirac equation?

The Dirac Lagrangian reads: \begin{equation*} \mathcal{L} = \bar\psi(i\not\partial-m)\psi.\tag{1} \end{equation*} It's invariant under the transformation $\psi(x)\rightarrow e^{i\alpha}\psi(x)$. Now ...
IGY's user avatar
  • 1,853
1 vote
2 answers
614 views

Noether's theorem under arbitrary coordinate transformation

Noether's theorem states that every differentiable symmetry of the action of a physical system has a corresponding conservation law. Suppose our action is of the form $S = \int d^4x\, \mathcal{L}(\...
amilton moreira's user avatar
2 votes
1 answer
493 views

Subtlety in derivation of Noether's theorem by Di Francesco

In the book 'Conformal Field Theory' by Di Francesco et al, a derivation of Noether's theorem is demonstrated by imposing that, what I believe is said to be a more elegant approach, the parameter $\...
CAF's user avatar
  • 3,599
3 votes
1 answer
1k views

Variation of the Action under infinitesimal arbitrary transformations and Noether's Theorem

Let's consider an arbitrary infinitesimal transformation of the fields and their coordinates : $$x'^{\mu}= x^{\mu} + \delta x^{\mu} = x^{\mu} + \frac{\delta x^{\mu}}{\delta{\omega}^a}{\omega}^a\tag{1}...
Alessandro's user avatar

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